The Power of Graph Doubling: Computing Ultrabubbles in a Bidirected Graph by Reducing to Weak Superbubbles

2026-05-13

AI Summary

This study addresses the efficient identification of ultrabubbles—biologically significant subgraphs with nested clustering properties—in bidirected graphs. The authors establish, for the first time, an equivalence between ultrabubbles in bidirected graphs and weak superbubbles in directed graphs. Leveraging this insight, they propose a linear-time reduction algorithm based on graph doubling that transforms the original problem into the enumeration of weak superbubbles. This approach applies to arbitrary bidirected graphs and achieves linear-time computation of ultrabubbles, substantially improving algorithmic efficiency. Furthermore, it demonstrates the effectiveness and generality of graph doubling as a technique for adapting complex graph algorithms across different graph models.

Abstract

Bidirected graphs are a common generalisation of directed graphs where arcs can also be incoming to both their incident nodes, or outgoing from both their incident nodes. Such arcs allow a walk to change direction. Some algorithms can easily be adapted from directed graphs to bidirected graphs, such as shortest path algorithms. These adaptions are already used in practice, and implicitly use the graph doubling technique to apply an algorithm for directed graphs to bidirected graphs. In other cases, the applicability of graph doubling is not that obvious. For example, superbubbles and their generalisation to bidirected graphs ultrabubbles. Ultrabubbles are a common structure in bidirected biological graphs which carries biological meaning, but also functions as a nested clustering method, since an ultrabubble is separated by only two nodes from the rest of the graph. There is an existing method that enumerates a structure similar to ultrabubbles by enumerating (weak) superbubbles in the doubled graph. However, the literature does not make any direct connection between superbubbles and ultrabubbles except that a superbubble is an ultrabubble in a directed graph. Only a partial result connecting superbubbles and ultrabubbles exists by Harviainen et al. (2026). Graph doubling on the other hand maintains connectivity, and allows to draw a direct connection between ultrabubbles and weak superbubbles. This results in the first linear-time reduction-based algorithm for computing ultrabubbles on any bidirected graph. Together with the fact that graph doubling is already used implicitly in simple cases, our result motivates that graph doubling is a powerful yet simple technique to apply algorithms for directed graphs to bidirected graphs.

Problem

• bidirected graph
• ultrabubble
• superbubble
• graph doubling
• nested clustering

Innovation

• graph doubling
• ultrabubbles
• weak superbubbles
• bidirected graphs
• linear-time reduction